Dividing Fractions Worksheets with Answer Key - Free Printable

To solve the problems on the "Dividing Fractions Worksheet," we need to follow these steps:

1. Convert mixed numbers to improper fractions (if necessary).
2. Rewrite the division as multiplication by the reciprocal of the divisor.
3. Simplify the resulting expression.

Let's solve each problem step by step.

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Problem 1: \( 3 \frac{2}{3} \div 6 \frac{1}{6} \)

#### Step 1: Convert mixed numbers to improper fractions
- \( 3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \)
- \( 6 \frac{1}{6} = \frac{6 \times 6 + 1}{6} = \frac{36 + 1}{6} = \frac{37}{6} \)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{11}{3} \div \frac{37}{6} = \frac{11}{3} \times \frac{6}{37} \]

#### Step 3: Simplify
\[ \frac{11}{3} \times \frac{6}{37} = \frac{11 \times 6}{3 \times 37} = \frac{66}{111} \]

#### Step 4: Simplify the fraction
The greatest common divisor (GCD) of 66 and 111 is 3.
\[ \frac{66 \div 3}{111 \div 3} = \frac{22}{37} \]

Answer:
\[ \boxed{\frac{22}{37}} \]

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Problem 2: \( 4 \frac{5}{6} \div 2 \frac{1}{3} \)

#### Step 1: Convert mixed numbers to improper fractions
- \( 4 \frac{5}{6} = \frac{4 \times 6 + 5}{6} = \frac{24 + 5}{6} = \frac{29}{6} \)
- \( 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{29}{6} \div \frac{7}{3} = \frac{29}{6} \times \frac{3}{7} \]

#### Step 3: Simplify
\[ \frac{29}{6} \times \frac{3}{7} = \frac{29 \times 3}{6 \times 7} = \frac{87}{42} \]

#### Step 4: Simplify the fraction
The GCD of 87 and 42 is 3.
\[ \frac{87 \div 3}{42 \div 3} = \frac{29}{14} \]

Answer:
\[ \boxed{\frac{29}{14}} \]

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Problem 3: \( 7 \frac{1}{3} \div \frac{5}{6} \)

#### Step 1: Convert mixed number to an improper fraction
- \( 7 \frac{1}{3} = \frac{7 \times 3 + 1}{3} = \frac{21 + 1}{3} = \frac{22}{3} \)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{22}{3} \div \frac{5}{6} = \frac{22}{3} \times \frac{6}{5} \]

#### Step 3: Simplify
\[ \frac{22}{3} \times \frac{6}{5} = \frac{22 \times 6}{3 \times 5} = \frac{132}{15} \]

#### Step 4: Simplify the fraction
The GCD of 132 and 15 is 3.
\[ \frac{132 \div 3}{15 \div 3} = \frac{44}{5} \]

Answer:
\[ \boxed{\frac{44}{5}} \]

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Problem 4: \( 4 \frac{3}{8} \div \frac{14}{16} \)

#### Step 1: Convert mixed number to an improper fraction
- \( 4 \frac{3}{8} = \frac{4 \times 8 + 3}{8} = \frac{32 + 3}{8} = \frac{35}{8} \)

#### Step 2: Simplify the divisor
\[ \frac{14}{16} = \frac{7}{8} \] (since the GCD of 14 and 16 is 2)

#### Step 3: Rewrite division as multiplication by the reciprocal
\[ \frac{35}{8} \div \frac{7}{8} = \frac{35}{8} \times \frac{8}{7} \]

#### Step 4: Simplify
\[ \frac{35}{8} \times \frac{8}{7} = \frac{35 \times 8}{8 \times 7} = \frac{35}{7} = 5 \]

Answer:
\[ \boxed{5} \]

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Problem 5: \( \frac{8}{9} \div \frac{72}{84} \)

#### Step 1: Simplify the divisor
\[ \frac{72}{84} = \frac{6}{7} \] (since the GCD of 72 and 84 is 12)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{8}{9} \div \frac{6}{7} = \frac{8}{9} \times \frac{7}{6} \]

#### Step 3: Simplify
\[ \frac{8}{9} \times \frac{7}{6} = \frac{8 \times 7}{9 \times 6} = \frac{56}{54} \]

#### Step 4: Simplify the fraction
The GCD of 56 and 54 is 2.
\[ \frac{56 \div 2}{54 \div 2} = \frac{28}{27} \]

Answer:
\[ \boxed{\frac{28}{27}} \]

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Problem 6: \( \frac{15}{16} \div \frac{20}{21} \)

#### Step 1: Rewrite division as multiplication by the reciprocal
\[ \frac{15}{16} \div \frac{20}{21} = \frac{15}{16} \times \frac{21}{20} \]

#### Step 2: Simplify
\[ \frac{15}{16} \times \frac{21}{20} = \frac{15 \times 21}{16 \times 20} = \frac{315}{320} \]

#### Step 3: Simplify the fraction
The GCD of 315 and 320 is 5.
\[ \frac{315 \div 5}{320 \div 5} = \frac{63}{64} \]

Answer:
\[ \boxed{\frac{63}{64}} \]

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Problem 7: \( \frac{11}{16} \div \frac{4}{5} \)

#### Step 1: Rewrite division as multiplication by the reciprocal
\[ \frac{11}{16} \div \frac{4}{5} = \frac{11}{16} \times \frac{5}{4} \]

#### Step 2: Simplify
\[ \frac{11}{16} \times \frac{5}{4} = \frac{11 \times 5}{16 \times 4} = \frac{55}{64} \]

Answer:
\[ \boxed{\frac{55}{64}} \]

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Problem 8: \( 8 \frac{2}{3} \div 2 \frac{1}{3} \)

#### Step 1: Convert mixed numbers to improper fractions
- \( 8 \frac{2}{3} = \frac{8 \times 3 + 2}{3} = \frac{24 + 2}{3} = \frac{26}{3} \)
- \( 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{26}{3} \div \frac{7}{3} = \frac{26}{3} \times \frac{3}{7} \]

#### Step 3: Simplify
\[ \frac{26}{3} \times \frac{3}{7} = \frac{26 \times 3}{3 \times 7} = \frac{26}{7} \]

Answer:
\[ \boxed{\frac{26}{7}} \]

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Problem 9: \( \frac{29}{5} \div 3 \frac{2}{9} \)

#### Step 1: Convert mixed number to an improper fraction
- \( 3 \frac{2}{9} = \frac{3 \times 9 + 2}{9} = \frac{27 + 2}{9} = \frac{29}{9} \)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{29}{5} \div \frac{29}{9} = \frac{29}{5} \times \frac{9}{29} \]

#### Step 3: Simplify
\[ \frac{29}{5} \times \frac{9}{29} = \frac{29 \times 9}{5 \times 29} = \frac{9}{5} \]

Answer:
\[ \boxed{\frac{9}{5}} \]

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Problem 10: \( \frac{3}{4} \div 7 \frac{3}{8} \)

#### Step 1: Convert mixed number to an improper fraction
- \( 7 \frac{3}{8} = \frac{7 \times 8 + 3}{8} = \frac{56 + 3}{8} = \frac{59}{8} \)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{3}{4} \div \frac{59}{8} = \frac{3}{4} \times \frac{8}{59} \]

#### Step 3: Simplify
\[ \frac{3}{4} \times \frac{8}{59} = \frac{3 \times 8}{4 \times 59} = \frac{24}{236} \]

#### Step 4: Simplify the fraction
The GCD of 24 and 236 is 4.
\[ \frac{24 \div 4}{236 \div 4} = \frac{6}{59} \]

Answer:
\[ \boxed{\frac{6}{59}} \]

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Problem 11: \( \frac{17}{2} \div \frac{14}{4} \)

#### Step 1: Simplify the divisor
\[ \frac{14}{4} = \frac{7}{2} \] (since the GCD of 14 and 4 is 2)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{17}{2} \div \frac{7}{2} = \frac{17}{2} \times \frac{2}{7} \]

#### Step 3: Simplify
\[ \frac{17}{2} \times \frac{2}{7} = \frac{17 \times 2}{2 \times 7} = \frac{17}{7} \]

Answer:
\[ \boxed{\frac{17}{7}} \]

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Problem 12: \( \frac{17}{3} \div \frac{34}{4} \)

#### Step 1: Simplify the divisor
\[ \frac{34}{4} = \frac{17}{2} \] (since the GCD of 34 and 4 is 2)

#### Step 2: Rewrite division as multiplication by the reciprocal
\[ \frac{17}{3} \div \frac{17}{2} = \frac{17}{3} \times \frac{2}{17} \]

#### Step 3: Simplify
\[ \frac{17}{3} \times \frac{2}{17} = \frac{17 \times 2}{3 \times 17} = \frac{2}{3} \]

Answer:
\[ \boxed{\frac{2}{3}} \]

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Problem 13: \( 6 \frac{1}{2} \div \frac{34}{4} \)

#### Step 1: Convert mixed number to an improper fraction
- \( 6 \frac{1}{2} = \frac{6 \times 2 + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2} \)

#### Step 2: Simplify the divisor
\[ \frac{34}{4} = \frac{17}{2} \] (since the GCD of 34 and 4 is 2)

#### Step 3: Rewrite division as multiplication by the reciprocal
\[ \frac{13}{2} \div \frac{17}{2} = \frac{13}{2} \times \frac{2}{17} \]

#### Step 4: Simplify
\[ \frac{13}{2} \times \frac{2}{17} = \frac{13 \times 2}{2 \times 17} = \frac{13}{17} \]

Answer:
\[ \boxed{\frac{13}{17}} \]

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Problem 14: \( \frac{24}{3} \div 6 \frac{2}{5} \)

#### Step 1: Simplify the dividend
\[ \frac{24}{3} = 8 \]

#### Step 2: Convert mixed number to an improper fraction
- \( 6 \frac{2}{5} = \frac{6 \times 5 + 2}{5} = \frac{30 + 2}{5} = \frac{32}{5} \)

#### Step 3: Rewrite division as multiplication by the reciprocal
\[ 8 \div \frac{32}{5} = 8 \times \frac{5}{32} \]

#### Step 4: Simplify
\[ 8 \times \frac{5}{32} = \frac{8 \times 5}{32} = \frac{40}{32} \]

#### Step 5: Simplify the fraction
The GCD of 40 and 32 is 8.
\[ \frac{40 \div 8}{32 \div 8} = \frac{5}{4} \]

Answer:
\[ \boxed{\frac{5}{4}} \]

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & \frac{22}{37} \\
2. & \frac{29}{14} \\
3. & \frac{44}{5} \\
4. & 5 \\
5. & \frac{28}{27} \\
6. & \frac{63}{64} \\
7. & \frac{55}{64} \\
8. & \frac{26}{7} \\
9. & \frac{9}{5} \\
10. & \frac{6}{59} \\
11. & \frac{17}{7} \\
12. & \frac{2}{3} \\
13. & \frac{13}{17} \\
14. & \frac{5}{4}
\end{array}
}
\]